Will I be held responsible for any debts that my parent incurs after they die?

You won't be responsible, but your parents estate will be responsible for claims of creditors who are owqed money for services or purchases by your parents.

Children do not inherit debt. However, this does not mean the your parent's creditor has no recourse. When a person dies, their estate is put into probate (whether or not they died with a Will). Probate allows for the distribution of assets, not debts. However, before any assets can so distributed to beneficiaries/heirs, all known debts must be paid by the executor/personal representative. Accordingly, the executor/PR will sell assets in the estate to pay for any debts that remain. Only after the debts are paid will the remaining assets be distributed. If there is no estate assets, then the debt extinguishes as a matter of law since there is nothing to collect against.